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1.1 Theory
An optical system's finite resolution is described by its optical transfer
function or OTF

(1) 
This represents the signal contribution of a point located an offset
from the point the system is measuring. For a perfect system,
,
where
is the DiracDelta function. However, for real
systems of finite resolution,
can be measured experimentally
by recording the image of a very small, pointlike source,
acting essentially as a baseline signal. The OTF is then given by
where is the (known) position of the pointlike source.
In addition, as the OTF depends only on the physical nature of the instruments,
it can be analysed to give a theoretical OTF for the system. This is as good
as the OTF obtained experimentally and as shown in Section 2.1.2
has several practical advantages over an experimental OTF.
Equation 1 has been defined in terms of
because this means that any measured image derives from the
original (``true'') image
by convolution with the optical
transfer function.

(2) 
Now when working in Fourier space, we can use the identity that convolution
is equivalent to multiplication of Fourier transforms to get

(3) 
where the boldface indicates a Fourier transform and the
Fourier transform of the optical transfer function is sometimes
referred to as the point spread function or psf. Quantitatively
sharpening the image to obtain
involves
the deconvolution of equation 2, which now
becomes the inversion of equation 3

(4) 
Unfortunately, real optical systems also suffer from the addition of noise by
the instruments involved in the system. We modify equation 2
to account for the noise due to a photon and the optical
instrument noise and obtain
for the collected noise term
. However,
cannot be known and so calculating Equation 5
is no longer possible. Further, considering small in Equation
4 we see that neglecting
will introduce major disturbances and artifacts in the restored image.
Subsections
Next: 1.1.1 Exact image enhancement
Up: 1 Introduction
Previous: 1 Introduction
Kevin Pulo
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